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Theorem bnj96 28897
Description: Technical lemma for bnj150 28908. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj96.1  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
Assertion
Ref Expression
bnj96  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  dom  F  =  1o )
Distinct variable groups:    x, A    x, R
Allowed substitution hint:    F( x)

Proof of Theorem bnj96
StepHypRef Expression
1 bnj93 28895 . . 3  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R )  e.  _V )
2 dmsnopg 5144 . . 3  |-  (  pred ( x ,  A ,  R )  e.  _V  ->  dom  { <. (/) ,  pred ( x ,  A ,  R ) >. }  =  { (/) } )
31, 2syl 15 . 2  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  dom  { <. (/) ,  pred ( x ,  A ,  R ) >. }  =  { (/) } )
4 bnj96.1 . . 3  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
54dmeqi 4880 . 2  |-  dom  F  =  dom  { <. (/) ,  pred ( x ,  A ,  R ) >. }
6 df1o2 6491 . 2  |-  1o  =  { (/) }
73, 5, 63eqtr4g 2340 1  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  dom  F  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   {csn 3640   <.cop 3643   dom cdm 4689   1oc1o 6472    predc-bnj14 28713    FrSe w-bnj15 28717
This theorem is referenced by:  bnj150  28908
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-suc 4398  df-dm 4699  df-1o 6479  df-bnj13 28716  df-bnj15 28718
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